%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : NUM414+1 : TPTP v9.0.0. Released v3.2.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n008.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:20:54 AM UTC 2025
% Result : Theorem 0.18s 0.37s
% Output : Proof 0.18s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.11 % Problem : NUM414+1 : TPTP v9.0.0. Released v3.2.0.
% 0.03/0.11 % Command : run_E %s %d THM
% 0.11/0.32 % Computer : n008.cluster.edu
% 0.11/0.32 % Model : x86_64 x86_64
% 0.11/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.32 % Memory : 8042.1875MB
% 0.11/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.32 % CPULimit : 300
% 0.11/0.32 % WCLimit : 300
% 0.11/0.32 % DateTime : Fri Jun 20 01:02:48 EDT 2025
% 0.11/0.32 % CPUTime :
% 0.18/0.37 % SZS status Theorem
% 0.18/0.37 % SZS output start Proof
% 0.18/0.37 tff(subset_type, type, (
% 0.18/0.37 subset: ( $i * $i ) > $o)).
% 0.18/0.37 tff(tptp_fun_A_15_type, type, (
% 0.18/0.37 tptp_fun_A_15: $i)).
% 0.18/0.37 tff(tptp_fun_B_16_type, type, (
% 0.18/0.37 tptp_fun_B_16: $i)).
% 0.18/0.37 tff(ordinal_subset_type, type, (
% 0.18/0.37 ordinal_subset: ( $i * $i ) > $o)).
% 0.18/0.37 tff(ordinal_type, type, (
% 0.18/0.37 ordinal: $i > $o)).
% 0.18/0.37 tff(proper_subset_type, type, (
% 0.18/0.37 proper_subset: ( $i * $i ) > $o)).
% 0.18/0.37 tff(1,plain,
% 0.18/0.37 ((~![A: $i] : ((~ordinal(A)) | ![B: $i] : ((~ordinal(B)) | (~((~proper_subset(A, B)) & (~(A = B)) & (~proper_subset(B, A))))))) <=> (~![A: $i] : ((~ordinal(A)) | ![B: $i] : ((~ordinal(B)) | (~((~proper_subset(A, B)) & (~(A = B)) & (~proper_subset(B, A)))))))),
% 0.18/0.37 inference(rewrite,[status(thm)],[])).
% 0.18/0.37 tff(2,plain,
% 0.18/0.37 ((~![A: $i] : (ordinal(A) => ![B: $i] : (ordinal(B) => (~(((~proper_subset(A, B)) & (~(A = B))) & (~proper_subset(B, A))))))) <=> (~![A: $i] : ((~ordinal(A)) | ![B: $i] : ((~ordinal(B)) | (~((~proper_subset(A, B)) & (~(A = B)) & (~proper_subset(B, A)))))))),
% 0.18/0.37 inference(rewrite,[status(thm)],[])).
% 0.18/0.37 tff(3,axiom,(~![A: $i] : (ordinal(A) => ![B: $i] : (ordinal(B) => (~(((~proper_subset(A, B)) & (~(A = B))) & (~proper_subset(B, A))))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t50_ordinal1')).
% 0.18/0.37 tff(4,plain,
% 0.18/0.37 (~![A: $i] : ((~ordinal(A)) | ![B: $i] : ((~ordinal(B)) | (~((~proper_subset(A, B)) & (~(A = B)) & (~proper_subset(B, A))))))),
% 0.18/0.37 inference(modus_ponens,[status(thm)],[3, 2])).
% 0.18/0.37 tff(5,plain,
% 0.18/0.37 (~![A: $i] : ((~ordinal(A)) | ![B: $i] : ((~ordinal(B)) | (~((~proper_subset(A, B)) & (~(A = B)) & (~proper_subset(B, A))))))),
% 0.18/0.37 inference(modus_ponens,[status(thm)],[4, 1])).
% 0.18/0.37 tff(6,plain,
% 0.18/0.37 (~![A: $i] : ((~ordinal(A)) | ![B: $i] : ((~ordinal(B)) | (~((~proper_subset(A, B)) & (~(A = B)) & (~proper_subset(B, A))))))),
% 0.18/0.37 inference(modus_ponens,[status(thm)],[5, 1])).
% 0.18/0.37 tff(7,plain,
% 0.18/0.37 (~![A: $i] : ((~ordinal(A)) | ![B: $i] : ((~ordinal(B)) | (~((~proper_subset(A, B)) & (~(A = B)) & (~proper_subset(B, A))))))),
% 0.18/0.37 inference(modus_ponens,[status(thm)],[6, 1])).
% 0.18/0.37 tff(8,plain,
% 0.18/0.37 (~((~ordinal(B!16)) | (~((~proper_subset(A!15, B!16)) & (~(A!15 = B!16)) & (~proper_subset(B!16, A!15)))))),
% 0.18/0.37 inference(and_elim,[status(thm)],[7])).
% 0.18/0.37 tff(9,plain,
% 0.18/0.37 (ordinal(B!16)),
% 0.18/0.37 inference(or_elim,[status(thm)],[8])).
% 0.18/0.37 tff(10,plain,
% 0.18/0.37 (ordinal(A!15)),
% 0.18/0.37 inference(and_elim,[status(thm)],[7])).
% 0.18/0.37 tff(11,plain,
% 0.18/0.37 (^[A: $i, B: $i] : refl(((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A))) <=> ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A))))),
% 0.18/0.37 inference(bind,[status(th)],[])).
% 0.18/0.37 tff(12,plain,
% 0.18/0.37 (![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A))) <=> ![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))),
% 0.18/0.37 inference(quant_intro,[status(thm)],[11])).
% 0.18/0.37 tff(13,plain,
% 0.18/0.37 (^[A: $i, B: $i] : trans(monotonicity(trans(monotonicity(rewrite((ordinal(A) & ordinal(B)) <=> (~((~ordinal(B)) | (~ordinal(A))))), ((~(ordinal(A) & ordinal(B))) <=> (~(~((~ordinal(B)) | (~ordinal(A))))))), rewrite((~(~((~ordinal(B)) | (~ordinal(A))))) <=> ((~ordinal(B)) | (~ordinal(A)))), ((~(ordinal(A) & ordinal(B))) <=> ((~ordinal(B)) | (~ordinal(A))))), (((~(ordinal(A) & ordinal(B))) | (ordinal_subset(A, B) <=> subset(A, B))) <=> (((~ordinal(B)) | (~ordinal(A))) | (ordinal_subset(A, B) <=> subset(A, B))))), rewrite((((~ordinal(B)) | (~ordinal(A))) | (ordinal_subset(A, B) <=> subset(A, B))) <=> ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))), (((~(ordinal(A) & ordinal(B))) | (ordinal_subset(A, B) <=> subset(A, B))) <=> ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))))),
% 0.18/0.37 inference(bind,[status(th)],[])).
% 0.18/0.37 tff(14,plain,
% 0.18/0.37 (![A: $i, B: $i] : ((~(ordinal(A) & ordinal(B))) | (ordinal_subset(A, B) <=> subset(A, B))) <=> ![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))),
% 0.18/0.37 inference(quant_intro,[status(thm)],[13])).
% 0.18/0.37 tff(15,plain,
% 0.18/0.37 (![A: $i, B: $i] : ((~(ordinal(A) & ordinal(B))) | (ordinal_subset(A, B) <=> subset(A, B))) <=> ![A: $i, B: $i] : ((~(ordinal(A) & ordinal(B))) | (ordinal_subset(A, B) <=> subset(A, B)))),
% 0.18/0.37 inference(rewrite,[status(thm)],[])).
% 0.18/0.37 tff(16,plain,
% 0.18/0.37 (^[A: $i, B: $i] : rewrite(((ordinal(A) & ordinal(B)) => (ordinal_subset(A, B) <=> subset(A, B))) <=> ((~(ordinal(A) & ordinal(B))) | (ordinal_subset(A, B) <=> subset(A, B))))),
% 0.18/0.37 inference(bind,[status(th)],[])).
% 0.18/0.37 tff(17,plain,
% 0.18/0.37 (![A: $i, B: $i] : ((ordinal(A) & ordinal(B)) => (ordinal_subset(A, B) <=> subset(A, B))) <=> ![A: $i, B: $i] : ((~(ordinal(A) & ordinal(B))) | (ordinal_subset(A, B) <=> subset(A, B)))),
% 0.18/0.37 inference(quant_intro,[status(thm)],[16])).
% 0.18/0.37 tff(18,axiom,(![A: $i, B: $i] : ((ordinal(A) & ordinal(B)) => (ordinal_subset(A, B) <=> subset(A, B)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','redefinition_r1_ordinal1')).
% 0.18/0.37 tff(19,plain,
% 0.18/0.37 (![A: $i, B: $i] : ((~(ordinal(A) & ordinal(B))) | (ordinal_subset(A, B) <=> subset(A, B)))),
% 0.18/0.37 inference(modus_ponens,[status(thm)],[18, 17])).
% 0.18/0.37 tff(20,plain,
% 0.18/0.37 (![A: $i, B: $i] : ((~(ordinal(A) & ordinal(B))) | (ordinal_subset(A, B) <=> subset(A, B)))),
% 0.18/0.37 inference(modus_ponens,[status(thm)],[19, 15])).
% 0.18/0.37 tff(21,plain,(
% 0.18/0.37 ![A: $i, B: $i] : ((~(ordinal(A) & ordinal(B))) | (ordinal_subset(A, B) <=> subset(A, B)))),
% 0.18/0.37 inference(skolemize,[status(sab)],[20])).
% 0.18/0.37 tff(22,plain,
% 0.18/0.37 (![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))),
% 0.18/0.37 inference(modus_ponens,[status(thm)],[21, 14])).
% 0.18/0.37 tff(23,plain,
% 0.18/0.37 (![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))),
% 0.18/0.37 inference(modus_ponens,[status(thm)],[22, 12])).
% 0.18/0.37 tff(24,plain,
% 0.18/0.37 (((~![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))) | ((~ordinal(A!15)) | (~ordinal(B!16)) | (ordinal_subset(B!16, A!15) <=> subset(B!16, A!15)))) <=> ((~![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))) | (~ordinal(A!15)) | (~ordinal(B!16)) | (ordinal_subset(B!16, A!15) <=> subset(B!16, A!15)))),
% 0.18/0.37 inference(rewrite,[status(thm)],[])).
% 0.18/0.37 tff(25,plain,
% 0.18/0.37 (((ordinal_subset(B!16, A!15) <=> subset(B!16, A!15)) | (~ordinal(A!15)) | (~ordinal(B!16))) <=> ((~ordinal(A!15)) | (~ordinal(B!16)) | (ordinal_subset(B!16, A!15) <=> subset(B!16, A!15)))),
% 0.18/0.37 inference(rewrite,[status(thm)],[])).
% 0.18/0.37 tff(26,plain,
% 0.18/0.37 (((~![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))) | ((ordinal_subset(B!16, A!15) <=> subset(B!16, A!15)) | (~ordinal(A!15)) | (~ordinal(B!16)))) <=> ((~![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))) | ((~ordinal(A!15)) | (~ordinal(B!16)) | (ordinal_subset(B!16, A!15) <=> subset(B!16, A!15))))),
% 0.18/0.37 inference(monotonicity,[status(thm)],[25])).
% 0.18/0.37 tff(27,plain,
% 0.18/0.37 (((~![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))) | ((ordinal_subset(B!16, A!15) <=> subset(B!16, A!15)) | (~ordinal(A!15)) | (~ordinal(B!16)))) <=> ((~![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))) | (~ordinal(A!15)) | (~ordinal(B!16)) | (ordinal_subset(B!16, A!15) <=> subset(B!16, A!15)))),
% 0.18/0.37 inference(transitivity,[status(thm)],[26, 24])).
% 0.18/0.37 tff(28,plain,
% 0.18/0.37 ((~![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))) | ((ordinal_subset(B!16, A!15) <=> subset(B!16, A!15)) | (~ordinal(A!15)) | (~ordinal(B!16)))),
% 0.18/0.37 inference(quant_inst,[status(thm)],[])).
% 0.18/0.37 tff(29,plain,
% 0.18/0.37 ((~![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))) | (~ordinal(A!15)) | (~ordinal(B!16)) | (ordinal_subset(B!16, A!15) <=> subset(B!16, A!15))),
% 0.18/0.37 inference(modus_ponens,[status(thm)],[28, 27])).
% 0.18/0.37 tff(30,plain,
% 0.18/0.37 (ordinal_subset(B!16, A!15) <=> subset(B!16, A!15)),
% 0.18/0.37 inference(unit_resolution,[status(thm)],[29, 23, 10, 9])).
% 0.18/0.37 tff(31,plain,
% 0.18/0.37 (((~![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))) | ((~ordinal(A!15)) | (~ordinal(B!16)) | (ordinal_subset(A!15, B!16) <=> subset(A!15, B!16)))) <=> ((~![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))) | (~ordinal(A!15)) | (~ordinal(B!16)) | (ordinal_subset(A!15, B!16) <=> subset(A!15, B!16)))),
% 0.18/0.37 inference(rewrite,[status(thm)],[])).
% 0.18/0.37 tff(32,plain,
% 0.18/0.37 (((ordinal_subset(A!15, B!16) <=> subset(A!15, B!16)) | (~ordinal(B!16)) | (~ordinal(A!15))) <=> ((~ordinal(A!15)) | (~ordinal(B!16)) | (ordinal_subset(A!15, B!16) <=> subset(A!15, B!16)))),
% 0.18/0.37 inference(rewrite,[status(thm)],[])).
% 0.18/0.37 tff(33,plain,
% 0.18/0.37 (((~![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))) | ((ordinal_subset(A!15, B!16) <=> subset(A!15, B!16)) | (~ordinal(B!16)) | (~ordinal(A!15)))) <=> ((~![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))) | ((~ordinal(A!15)) | (~ordinal(B!16)) | (ordinal_subset(A!15, B!16) <=> subset(A!15, B!16))))),
% 0.18/0.37 inference(monotonicity,[status(thm)],[32])).
% 0.18/0.37 tff(34,plain,
% 0.18/0.37 (((~![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))) | ((ordinal_subset(A!15, B!16) <=> subset(A!15, B!16)) | (~ordinal(B!16)) | (~ordinal(A!15)))) <=> ((~![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))) | (~ordinal(A!15)) | (~ordinal(B!16)) | (ordinal_subset(A!15, B!16) <=> subset(A!15, B!16)))),
% 0.18/0.37 inference(transitivity,[status(thm)],[33, 31])).
% 0.18/0.37 tff(35,plain,
% 0.18/0.37 ((~![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))) | ((ordinal_subset(A!15, B!16) <=> subset(A!15, B!16)) | (~ordinal(B!16)) | (~ordinal(A!15)))),
% 0.18/0.37 inference(quant_inst,[status(thm)],[])).
% 0.18/0.37 tff(36,plain,
% 0.18/0.37 ((~![A: $i, B: $i] : ((ordinal_subset(A, B) <=> subset(A, B)) | (~ordinal(B)) | (~ordinal(A)))) | (~ordinal(A!15)) | (~ordinal(B!16)) | (ordinal_subset(A!15, B!16) <=> subset(A!15, B!16))),
% 0.18/0.37 inference(modus_ponens,[status(thm)],[35, 34])).
% 0.18/0.37 tff(37,plain,
% 0.18/0.37 (ordinal_subset(A!15, B!16) <=> subset(A!15, B!16)),
% 0.18/0.37 inference(unit_resolution,[status(thm)],[36, 23, 10, 9])).
% 0.18/0.37 tff(38,plain,
% 0.18/0.37 (^[A: $i, B: $i] : refl((~(((~subset(A, B)) | (A = B)) <=> proper_subset(A, B))) <=> (~(((~subset(A, B)) | (A = B)) <=> proper_subset(A, B))))),
% 0.18/0.37 inference(bind,[status(th)],[])).
% 0.18/0.37 tff(39,plain,
% 0.18/0.37 (![A: $i, B: $i] : (~(((~subset(A, B)) | (A = B)) <=> proper_subset(A, B))) <=> ![A: $i, B: $i] : (~(((~subset(A, B)) | (A = B)) <=> proper_subset(A, B)))),
% 0.18/0.37 inference(quant_intro,[status(thm)],[38])).
% 0.18/0.37 tff(40,plain,
% 0.18/0.37 (^[A: $i, B: $i] : trans(trans(monotonicity(rewrite((subset(A, B) & (~(A = B))) <=> (~((~subset(A, B)) | (A = B)))), ((proper_subset(A, B) <=> (subset(A, B) & (~(A = B)))) <=> (proper_subset(A, B) <=> (~((~subset(A, B)) | (A = B)))))), rewrite((proper_subset(A, B) <=> (~((~subset(A, B)) | (A = B)))) <=> (~(((~subset(A, B)) | (A = B)) <=> proper_subset(A, B)))), ((proper_subset(A, B) <=> (subset(A, B) & (~(A = B)))) <=> (~(((~subset(A, B)) | (A = B)) <=> proper_subset(A, B))))), rewrite((~(((~subset(A, B)) | (A = B)) <=> proper_subset(A, B))) <=> (~(((~subset(A, B)) | (A = B)) <=> proper_subset(A, B)))), ((proper_subset(A, B) <=> (subset(A, B) & (~(A = B)))) <=> (~(((~subset(A, B)) | (A = B)) <=> proper_subset(A, B)))))),
% 0.18/0.37 inference(bind,[status(th)],[])).
% 0.18/0.37 tff(41,plain,
% 0.18/0.37 (![A: $i, B: $i] : (proper_subset(A, B) <=> (subset(A, B) & (~(A = B)))) <=> ![A: $i, B: $i] : (~(((~subset(A, B)) | (A = B)) <=> proper_subset(A, B)))),
% 0.18/0.37 inference(quant_intro,[status(thm)],[40])).
% 0.18/0.37 tff(42,plain,
% 0.18/0.37 (![A: $i, B: $i] : (proper_subset(A, B) <=> (subset(A, B) & (~(A = B)))) <=> ![A: $i, B: $i] : (proper_subset(A, B) <=> (subset(A, B) & (~(A = B))))),
% 0.18/0.37 inference(rewrite,[status(thm)],[])).
% 0.18/0.37 tff(43,axiom,(![A: $i, B: $i] : (proper_subset(A, B) <=> (subset(A, B) & (~(A = B))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','d8_xboole_0')).
% 0.18/0.37 tff(44,plain,
% 0.18/0.37 (![A: $i, B: $i] : (proper_subset(A, B) <=> (subset(A, B) & (~(A = B))))),
% 0.18/0.37 inference(modus_ponens,[status(thm)],[43, 42])).
% 0.18/0.37 tff(45,plain,(
% 0.18/0.37 ![A: $i, B: $i] : (proper_subset(A, B) <=> (subset(A, B) & (~(A = B))))),
% 0.18/0.37 inference(skolemize,[status(sab)],[44])).
% 0.18/0.37 tff(46,plain,
% 0.18/0.37 (![A: $i, B: $i] : (~(((~subset(A, B)) | (A = B)) <=> proper_subset(A, B)))),
% 0.18/0.37 inference(modus_ponens,[status(thm)],[45, 41])).
% 0.18/0.37 tff(47,plain,
% 0.18/0.37 (![A: $i, B: $i] : (~(((~subset(A, B)) | (A = B)) <=> proper_subset(A, B)))),
% 0.18/0.37 inference(modus_ponens,[status(thm)],[46, 39])).
% 0.18/0.37 tff(48,plain,
% 0.18/0.37 ((~![A: $i, B: $i] : (~(((~subset(A, B)) | (A = B)) <=> proper_subset(A, B)))) | (~(((~subset(A!15, B!16)) | (A!15 = B!16)) <=> proper_subset(A!15, B!16)))),
% 0.18/0.37 inference(quant_inst,[status(thm)],[])).
% 0.18/0.37 tff(49,plain,
% 0.18/0.37 (~(((~subset(A!15, B!16)) | (A!15 = B!16)) <=> proper_subset(A!15, B!16))),
% 0.18/0.37 inference(unit_resolution,[status(thm)],[48, 47])).
% 0.18/0.37 tff(50,plain,
% 0.18/0.37 ((~proper_subset(A!15, B!16)) & (~(A!15 = B!16)) & (~proper_subset(B!16, A!15))),
% 0.18/0.37 inference(or_elim,[status(thm)],[8])).
% 0.18/0.37 tff(51,plain,
% 0.18/0.37 (~proper_subset(A!15, B!16)),
% 0.18/0.37 inference(and_elim,[status(thm)],[50])).
% 0.18/0.37 tff(52,plain,
% 0.18/0.37 ((((~subset(A!15, B!16)) | (A!15 = B!16)) <=> proper_subset(A!15, B!16)) | ((~subset(A!15, B!16)) | (A!15 = B!16)) | proper_subset(A!15, B!16)),
% 0.18/0.37 inference(tautology,[status(thm)],[])).
% 0.18/0.37 tff(53,plain,
% 0.18/0.37 ((((~subset(A!15, B!16)) | (A!15 = B!16)) <=> proper_subset(A!15, B!16)) | ((~subset(A!15, B!16)) | (A!15 = B!16))),
% 0.18/0.37 inference(unit_resolution,[status(thm)],[52, 51])).
% 0.18/0.37 tff(54,plain,
% 0.18/0.37 ((~subset(A!15, B!16)) | (A!15 = B!16)),
% 0.18/0.37 inference(unit_resolution,[status(thm)],[53, 49])).
% 0.18/0.37 tff(55,plain,
% 0.18/0.37 (~(A!15 = B!16)),
% 0.18/0.37 inference(and_elim,[status(thm)],[50])).
% 0.18/0.37 tff(56,plain,
% 0.18/0.37 ((~((~subset(A!15, B!16)) | (A!15 = B!16))) | (~subset(A!15, B!16)) | (A!15 = B!16)),
% 0.18/0.37 inference(tautology,[status(thm)],[])).
% 0.18/0.37 tff(57,plain,
% 0.18/0.37 ((~((~subset(A!15, B!16)) | (A!15 = B!16))) | (~subset(A!15, B!16))),
% 0.18/0.37 inference(unit_resolution,[status(thm)],[56, 55])).
% 0.18/0.37 tff(58,plain,
% 0.18/0.37 (~subset(A!15, B!16)),
% 0.18/0.37 inference(unit_resolution,[status(thm)],[57, 54])).
% 0.18/0.37 tff(59,plain,
% 0.18/0.37 ((~(ordinal_subset(A!15, B!16) <=> subset(A!15, B!16))) | (~ordinal_subset(A!15, B!16)) | subset(A!15, B!16)),
% 0.18/0.37 inference(tautology,[status(thm)],[])).
% 0.18/0.37 tff(60,plain,
% 0.18/0.37 ((~(ordinal_subset(A!15, B!16) <=> subset(A!15, B!16))) | (~ordinal_subset(A!15, B!16))),
% 0.18/0.37 inference(unit_resolution,[status(thm)],[59, 58])).
% 0.18/0.37 tff(61,plain,
% 0.18/0.37 (~ordinal_subset(A!15, B!16)),
% 0.18/0.37 inference(unit_resolution,[status(thm)],[60, 37])).
% 0.18/0.37 tff(62,plain,
% 0.18/0.37 (^[A: $i, B: $i] : refl((ordinal_subset(B, A) | ordinal_subset(A, B) | (~ordinal(B)) | (~ordinal(A))) <=> (ordinal_subset(B, A) | ordinal_subset(A, B) | (~ordinal(B)) | (~ordinal(A))))),
% 0.18/0.37 inference(bind,[status(th)],[])).
% 0.18/0.37 tff(63,plain,
% 0.18/0.37 (![A: $i, B: $i] : (ordinal_subset(B, A) | ordinal_subset(A, B) | (~ordinal(B)) | (~ordinal(A))) <=> ![A: $i, B: $i] : (ordinal_subset(B, A) | ordinal_subset(A, B) | (~ordinal(B)) | (~ordinal(A)))),
% 0.18/0.37 inference(quant_intro,[status(thm)],[62])).
% 0.18/0.37 tff(64,plain,
% 0.18/0.37 (^[A: $i, B: $i] : trans(monotonicity(trans(monotonicity(rewrite((ordinal(A) & ordinal(B)) <=> (~((~ordinal(B)) | (~ordinal(A))))), ((~(ordinal(A) & ordinal(B))) <=> (~(~((~ordinal(B)) | (~ordinal(A))))))), rewrite((~(~((~ordinal(B)) | (~ordinal(A))))) <=> ((~ordinal(B)) | (~ordinal(A)))), ((~(ordinal(A) & ordinal(B))) <=> ((~ordinal(B)) | (~ordinal(A))))), ((ordinal_subset(B, A) | ordinal_subset(A, B) | (~(ordinal(A) & ordinal(B)))) <=> (ordinal_subset(B, A) | ordinal_subset(A, B) | ((~ordinal(B)) | (~ordinal(A)))))), rewrite((ordinal_subset(B, A) | ordinal_subset(A, B) | ((~ordinal(B)) | (~ordinal(A)))) <=> (ordinal_subset(B, A) | ordinal_subset(A, B) | (~ordinal(B)) | (~ordinal(A)))), ((ordinal_subset(B, A) | ordinal_subset(A, B) | (~(ordinal(A) & ordinal(B)))) <=> (ordinal_subset(B, A) | ordinal_subset(A, B) | (~ordinal(B)) | (~ordinal(A)))))),
% 0.18/0.37 inference(bind,[status(th)],[])).
% 0.18/0.37 tff(65,plain,
% 0.18/0.37 (![A: $i, B: $i] : (ordinal_subset(B, A) | ordinal_subset(A, B) | (~(ordinal(A) & ordinal(B)))) <=> ![A: $i, B: $i] : (ordinal_subset(B, A) | ordinal_subset(A, B) | (~ordinal(B)) | (~ordinal(A)))),
% 0.18/0.37 inference(quant_intro,[status(thm)],[64])).
% 0.18/0.38 tff(66,plain,
% 0.18/0.38 (![A: $i, B: $i] : (ordinal_subset(B, A) | ordinal_subset(A, B) | (~(ordinal(A) & ordinal(B)))) <=> ![A: $i, B: $i] : (ordinal_subset(B, A) | ordinal_subset(A, B) | (~(ordinal(A) & ordinal(B))))),
% 0.18/0.38 inference(rewrite,[status(thm)],[])).
% 0.18/0.38 tff(67,plain,
% 0.18/0.38 (^[A: $i, B: $i] : trans(monotonicity(rewrite((ordinal_subset(A, B) | ordinal_subset(B, A)) <=> (ordinal_subset(B, A) | ordinal_subset(A, B))), (((ordinal(A) & ordinal(B)) => (ordinal_subset(A, B) | ordinal_subset(B, A))) <=> ((ordinal(A) & ordinal(B)) => (ordinal_subset(B, A) | ordinal_subset(A, B))))), rewrite(((ordinal(A) & ordinal(B)) => (ordinal_subset(B, A) | ordinal_subset(A, B))) <=> (ordinal_subset(B, A) | ordinal_subset(A, B) | (~(ordinal(A) & ordinal(B))))), (((ordinal(A) & ordinal(B)) => (ordinal_subset(A, B) | ordinal_subset(B, A))) <=> (ordinal_subset(B, A) | ordinal_subset(A, B) | (~(ordinal(A) & ordinal(B))))))),
% 0.18/0.38 inference(bind,[status(th)],[])).
% 0.18/0.38 tff(68,plain,
% 0.18/0.38 (![A: $i, B: $i] : ((ordinal(A) & ordinal(B)) => (ordinal_subset(A, B) | ordinal_subset(B, A))) <=> ![A: $i, B: $i] : (ordinal_subset(B, A) | ordinal_subset(A, B) | (~(ordinal(A) & ordinal(B))))),
% 0.18/0.38 inference(quant_intro,[status(thm)],[67])).
% 0.18/0.38 tff(69,axiom,(![A: $i, B: $i] : ((ordinal(A) & ordinal(B)) => (ordinal_subset(A, B) | ordinal_subset(B, A)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','connectedness_r1_ordinal1')).
% 0.18/0.38 tff(70,plain,
% 0.18/0.38 (![A: $i, B: $i] : (ordinal_subset(B, A) | ordinal_subset(A, B) | (~(ordinal(A) & ordinal(B))))),
% 0.18/0.38 inference(modus_ponens,[status(thm)],[69, 68])).
% 0.18/0.38 tff(71,plain,
% 0.18/0.38 (![A: $i, B: $i] : (ordinal_subset(B, A) | ordinal_subset(A, B) | (~(ordinal(A) & ordinal(B))))),
% 0.18/0.38 inference(modus_ponens,[status(thm)],[70, 66])).
% 0.18/0.38 tff(72,plain,(
% 0.18/0.38 ![A: $i, B: $i] : (ordinal_subset(B, A) | ordinal_subset(A, B) | (~(ordinal(A) & ordinal(B))))),
% 0.18/0.38 inference(skolemize,[status(sab)],[71])).
% 0.18/0.38 tff(73,plain,
% 0.18/0.38 (![A: $i, B: $i] : (ordinal_subset(B, A) | ordinal_subset(A, B) | (~ordinal(B)) | (~ordinal(A)))),
% 0.18/0.38 inference(modus_ponens,[status(thm)],[72, 65])).
% 0.18/0.38 tff(74,plain,
% 0.18/0.38 (![A: $i, B: $i] : (ordinal_subset(B, A) | ordinal_subset(A, B) | (~ordinal(B)) | (~ordinal(A)))),
% 0.18/0.38 inference(modus_ponens,[status(thm)],[73, 63])).
% 0.18/0.38 tff(75,plain,
% 0.18/0.38 (((~![A: $i, B: $i] : (ordinal_subset(B, A) | ordinal_subset(A, B) | (~ordinal(B)) | (~ordinal(A)))) | ((~ordinal(A!15)) | (~ordinal(B!16)) | ordinal_subset(A!15, B!16) | ordinal_subset(B!16, A!15))) <=> ((~![A: $i, B: $i] : (ordinal_subset(B, A) | ordinal_subset(A, B) | (~ordinal(B)) | (~ordinal(A)))) | (~ordinal(A!15)) | (~ordinal(B!16)) | ordinal_subset(A!15, B!16) | ordinal_subset(B!16, A!15))),
% 0.18/0.38 inference(rewrite,[status(thm)],[])).
% 0.18/0.38 tff(76,plain,
% 0.18/0.38 ((ordinal_subset(A!15, B!16) | ordinal_subset(B!16, A!15) | (~ordinal(A!15)) | (~ordinal(B!16))) <=> ((~ordinal(A!15)) | (~ordinal(B!16)) | ordinal_subset(A!15, B!16) | ordinal_subset(B!16, A!15))),
% 0.18/0.38 inference(rewrite,[status(thm)],[])).
% 0.18/0.38 tff(77,plain,
% 0.18/0.38 (((~![A: $i, B: $i] : (ordinal_subset(B, A) | ordinal_subset(A, B) | (~ordinal(B)) | (~ordinal(A)))) | (ordinal_subset(A!15, B!16) | ordinal_subset(B!16, A!15) | (~ordinal(A!15)) | (~ordinal(B!16)))) <=> ((~![A: $i, B: $i] : (ordinal_subset(B, A) | ordinal_subset(A, B) | (~ordinal(B)) | (~ordinal(A)))) | ((~ordinal(A!15)) | (~ordinal(B!16)) | ordinal_subset(A!15, B!16) | ordinal_subset(B!16, A!15)))),
% 0.18/0.38 inference(monotonicity,[status(thm)],[76])).
% 0.18/0.38 tff(78,plain,
% 0.18/0.38 (((~![A: $i, B: $i] : (ordinal_subset(B, A) | ordinal_subset(A, B) | (~ordinal(B)) | (~ordinal(A)))) | (ordinal_subset(A!15, B!16) | ordinal_subset(B!16, A!15) | (~ordinal(A!15)) | (~ordinal(B!16)))) <=> ((~![A: $i, B: $i] : (ordinal_subset(B, A) | ordinal_subset(A, B) | (~ordinal(B)) | (~ordinal(A)))) | (~ordinal(A!15)) | (~ordinal(B!16)) | ordinal_subset(A!15, B!16) | ordinal_subset(B!16, A!15))),
% 0.18/0.38 inference(transitivity,[status(thm)],[77, 75])).
% 0.18/0.38 tff(79,plain,
% 0.18/0.38 ((~![A: $i, B: $i] : (ordinal_subset(B, A) | ordinal_subset(A, B) | (~ordinal(B)) | (~ordinal(A)))) | (ordinal_subset(A!15, B!16) | ordinal_subset(B!16, A!15) | (~ordinal(A!15)) | (~ordinal(B!16)))),
% 0.18/0.38 inference(quant_inst,[status(thm)],[])).
% 0.18/0.38 tff(80,plain,
% 0.18/0.38 ((~![A: $i, B: $i] : (ordinal_subset(B, A) | ordinal_subset(A, B) | (~ordinal(B)) | (~ordinal(A)))) | (~ordinal(A!15)) | (~ordinal(B!16)) | ordinal_subset(A!15, B!16) | ordinal_subset(B!16, A!15)),
% 0.18/0.38 inference(modus_ponens,[status(thm)],[79, 78])).
% 0.18/0.38 tff(81,plain,
% 0.18/0.38 (ordinal_subset(B!16, A!15)),
% 0.18/0.38 inference(unit_resolution,[status(thm)],[80, 74, 10, 9, 61])).
% 0.18/0.38 tff(82,plain,
% 0.18/0.38 ((~![A: $i, B: $i] : (~(((~subset(A, B)) | (A = B)) <=> proper_subset(A, B)))) | (~(((~subset(B!16, A!15)) | (B!16 = A!15)) <=> proper_subset(B!16, A!15)))),
% 0.18/0.38 inference(quant_inst,[status(thm)],[])).
% 0.18/0.38 tff(83,plain,
% 0.18/0.38 (~(((~subset(B!16, A!15)) | (B!16 = A!15)) <=> proper_subset(B!16, A!15))),
% 0.18/0.38 inference(unit_resolution,[status(thm)],[82, 47])).
% 0.18/0.38 tff(84,plain,
% 0.18/0.38 (~proper_subset(B!16, A!15)),
% 0.18/0.38 inference(and_elim,[status(thm)],[50])).
% 0.18/0.38 tff(85,plain,
% 0.18/0.38 ((((~subset(B!16, A!15)) | (B!16 = A!15)) <=> proper_subset(B!16, A!15)) | ((~subset(B!16, A!15)) | (B!16 = A!15)) | proper_subset(B!16, A!15)),
% 0.18/0.38 inference(tautology,[status(thm)],[])).
% 0.18/0.38 tff(86,plain,
% 0.18/0.38 ((((~subset(B!16, A!15)) | (B!16 = A!15)) <=> proper_subset(B!16, A!15)) | ((~subset(B!16, A!15)) | (B!16 = A!15))),
% 0.18/0.38 inference(unit_resolution,[status(thm)],[85, 84])).
% 0.18/0.38 tff(87,plain,
% 0.18/0.38 ((~subset(B!16, A!15)) | (B!16 = A!15)),
% 0.18/0.38 inference(unit_resolution,[status(thm)],[86, 83])).
% 0.18/0.38 tff(88,plain,
% 0.18/0.38 ((B!16 = A!15) <=> (A!15 = B!16)),
% 0.18/0.38 inference(commutativity,[status(thm)],[])).
% 0.18/0.38 tff(89,plain,
% 0.18/0.38 ((A!15 = B!16) <=> (B!16 = A!15)),
% 0.18/0.38 inference(symmetry,[status(thm)],[88])).
% 0.18/0.38 tff(90,plain,
% 0.18/0.38 ((~(A!15 = B!16)) <=> (~(B!16 = A!15))),
% 0.18/0.38 inference(monotonicity,[status(thm)],[89])).
% 0.18/0.38 tff(91,plain,
% 0.18/0.38 (~(B!16 = A!15)),
% 0.18/0.38 inference(modus_ponens,[status(thm)],[55, 90])).
% 0.18/0.38 tff(92,plain,
% 0.18/0.38 ((~((~subset(B!16, A!15)) | (B!16 = A!15))) | (~subset(B!16, A!15)) | (B!16 = A!15)),
% 0.18/0.38 inference(tautology,[status(thm)],[])).
% 0.18/0.38 tff(93,plain,
% 0.18/0.38 (~subset(B!16, A!15)),
% 0.18/0.38 inference(unit_resolution,[status(thm)],[92, 91, 87])).
% 0.18/0.38 tff(94,plain,
% 0.18/0.38 ((~(ordinal_subset(B!16, A!15) <=> subset(B!16, A!15))) | (~ordinal_subset(B!16, A!15)) | subset(B!16, A!15)),
% 0.18/0.38 inference(tautology,[status(thm)],[])).
% 0.18/0.38 tff(95,plain,
% 0.18/0.38 ((~(ordinal_subset(B!16, A!15) <=> subset(B!16, A!15))) | (~ordinal_subset(B!16, A!15))),
% 0.18/0.38 inference(unit_resolution,[status(thm)],[94, 93])).
% 0.18/0.38 tff(96,plain,
% 0.18/0.38 ($false),
% 0.18/0.38 inference(unit_resolution,[status(thm)],[95, 81, 30])).
% 0.18/0.38 % SZS output end Proof
% 0.18/0.38 % E exiting
%------------------------------------------------------------------------------